Dynamic output feedback control for a walking assistance training robot to handle shifts in the center of gravity and time-varying arm of force in omniwheel

Abstract

In this study, we designed a dynamic output feedback control for a walking assistance training robot. The proposed system addresses the problems of shifts in the center of gravity and vibration associated with time-varying arm of force in omniwheel to improve the accuracy of the trajectory tracking of the walking assistance training robot. The dynamic output feedback controller was developed by constructing a velocity observer on a stochastic dynamic model of the walking assistance training robot via integration with a Lyapunov function. An analysis of the time-varying arm of force in omniwheel revealed that it increased the accuracy of the walking assistance training robot dynamic model. Thus, an adaptive law was designed such that the vibration caused by the time-varying arm of force in omniwheel was eliminated. The mean absolute practical stability in the position and velocity tracking errors was verified based on Young’s inequality and stochastic stability theory. The simulation results show that the walking assistance training robot with the shifts in the center of gravity was able to track a designed trajectory and that the application of the adaptive law effectively eliminates the vibrations caused by the time-varying arm of force in omniwheel.

Keywords

Walking assistance training robot shifts in the center of gravity time-varying arm of force in omniwheel velocity observer dynamic output feedback control

Introduction

Lower limb rehabilitation robots have become a focal point of modern medical technology due to population aging and rates of disability are in a trend of rising year by year.^1
–3 Lower limb rehabilitation robots can be divided into two types⁴ those that exercise a patient’s leg muscles without actual movement by maintaining a fixed posture, such as sitting,⁵ lying down,⁶ or hanging,⁷ and those that are mounted on motion systems, including exoskeleton robots^8,9 and mobile robots. The first type of lower limb rehabilitation robots is more suitable for patients with severe muscle injuries but do not help to recover neurological function. On the other hand, exoskeleton robots help the patient move by applying supporting forces externally to the legs and must maintain balance to ensure the safety of the patient, an area of active research. Due to the reduced ability of the elderly to maintain balance and lower velocity of motion, mobile robots are more applicable for the elderly. Mobile lower limb rehabilitation robots may also be applied in hospitals and homes, so they must be able to operate in narrow spaces; therefore, they should be mounted with mechanical structures that are capable of omnidirectional motion.

Standard mobile robots can just realize the motions of forward, backward, left, and right, but are not capable of lateral movement.¹⁰ Omniwheels (i.e. omnidirectional wheels) have been incorporated into mobile robots to provide them with three degrees of freedom so they can obtain the movement with any direction¹¹ and, thus, maneuver through narrow and complex areas with accurate positioning. For example, an omnidirectional collaborative robot can help its human user to drive a heavy load along a designated path.¹² In a previous study, an omnidirectional configuration and control method were presented for a miniature-forklift guided vehicle, which can be applied in logistics, production, and maintenance.¹³ Compared to the traditional cross-country vehicles, omnidirectional robots have superior maneuverability and cross-country capability.¹⁴

Therefore, in this study, four omniwheels were mounted on a walking assistance training robot to allow it to move omnidirectionally; however, this also presented a challenge in the control of the robot. Unlike vehicles with standard wheels, omnidirectional vehicles suffer from noticeable jitter in the orientation angle. Unfortunately, the majority of the former researches neglected the problem while others attempted to reduce this jitter by adjusting the gains in the controller. Here, we also investigated the time-varying arm of force in omniwheel to analyze the reason of the jitter in the orientation angle. Then, we applied an adaptive technique to eliminate this jitter.

It is well known that accurate position and velocity data are essential for mechanical control systems; this is especially true for the walking assistance training robot since it requires very precise trajectory tracking. Usually, the position of the walking assistance training robot can be detected by high-precision sensors, such as a camera, infrared sensor, or radar. However, high-precision mechanical sensors to measure velocity are expensive, very sensitive to changes in the external environment, easily perturbed by noise, and are large and heavy; consequently, they are often omitted. To overcome these problems, an output feedback controller that does not require velocity measurements has been developed by leveraging a velocity observer to estimate the velocity of the walking assistance training robot.

For the practical applications of mechanical systems, random disturbances and random uncertainties are inevitable but may greatly impact the mechanical performance and properties. In the past few decades, the stochastic stability theory has been widely used to address this problem with excellent results.^15

–19 Recently, the stochastic theory has been used to develop some effective control methods for real mechanical system applications. For example, a stochastic model was constructed for a manipulator system working in an environment with random vibrations.²⁰ In another study, an adaptive control approach was developed for a flexible-jointed robot faced with random disturbances²¹ Moreover, a stochastic model and an H∞-robust method were presented to handle the noise in the input–output measurements of a floating raft.²² However, these methods only consider the random vibration in the input channels (external disturbances). Another study²³ was conducted to investigate a symbolic and numeric scheme to solve linear differential equations with random parameter uncertainties. However, few results regarding nonlinear dynamic systems with the internal random parameters have been reported. In our previous study,²⁴ we proposed a stochastic dynamic model that considers random shifts in the center of gravity (i.e. an internal random parameter) and designed a robust adaptive tracking controller that stabilizes the trajectory-tracking error of the system using stochastic modeling.

Here, we propose a novel stochastic dynamic model to investigate further the problems associated with time-varying arm of force in omniwheel and random shifts in the center of gravity without the use of a velocity sensor. High-precision trajectory tracking is a challenging task for the walking assistance training robot model due to the random shifts in the center of gravity. Thus, the motion of the walking assistance training robot with random shifts in the center of gravity is depicted by a stochastic dynamic model by converting the random parameters into random disturbances. Moreover, by investigating the time-varying arm of force in omniwheel, we show the cause of the angle jitter and design an adaptive law to adjust the unknown vector in the controller and, thereby, eliminate the angle jitter. In addition, a velocity sensor is omitted from the walking assistance training robot design and, instead, a velocity observer was proposed to address the dynamic output feedback control and make the tracking error system exponentially practically stable. Then, it was verified that the tracking error and observation errors tend to an arbitrarily small neighborhood around zero upon turning the design parameters based on the stochastic stability theorem and Markov’s inequality. Finally, a stability analysis was conducted and simulations were executed to show that the adaptive law eliminates the angle jitter, the velocity observation is accurate, and the walking assistance training robot tracks a designed trajectory.

Improved stochastic dynamic model of walking assistance training robot with the random shifts in the center of gravity and time-varying arm of force in omniwheel

Figure 1 shows the structure of the walking assistance training robot.²⁵ During rehabilitation training, the walking assistance training robot works with the user and, thus, experiences random shifts in the center of gravity as shown in the coordinate system in Figure 2.

In Figure 2, $Σ (X, O, Y)$ and $Σ (X^{'}, C, Y^{'})$ are the absolute and translated Cartesian coordinate systems, respectively; v is the walking assistance training robot speed, which is oriented at an angle of α relative to the $X^{'}$ -axis; f_i $(i = 1, 2, 3, 4)$ is the ith omniwheel force; v_i represents the ith omniwheel speed; C is the geometric center; G is the center of gravity of the walking assistance training robot with the load of the user; r ₀ is the distance of a vector between C and G, which is oriented at an angle of $β$ relative to the $X^{'}$ -axis; $l_{i}$ is the distance from G to the ith omniwheel (this vector is oriented at an angle of $ϕ_{i}$ relative to the $X^{'}$ -axis); Lis the distance from C to each omniwheel; and $λ_{i}$ is the perpendicular distance between G and the trajectory of the ith omniwheel (where the trajectory is at an angle of $θ_{i}$ relative to the $X^{'}$ -axis).

To design the controller, the dynamic model is given as follows²⁵

M K (t) {\ddot{X}}_{r} (t) + M \dot{K} (t) {\dot{X}}_{r} (t) = B (t) u_{r} (t)

where

M = [\begin{matrix} m_{r} + m_{u} & 0 & 0 \\ 0 & m_{r} + m_{u} & 0 \\ 0 & 0 & I^{'} + m_{u} r_{0}^{2} \end{matrix}], K (t) = [\begin{matrix} 1 & 0 & p (t) \\ 0 & 1 & q (t) \\ 0 & 0 & 1 \end{matrix}],

\begin{array}{l} X_{r} (t) = [\begin{matrix} x_{r} (t) \\ y_{r} (t) \\ θ_{r} (t) \end{matrix}], \begin{matrix} p (t) = \frac{1}{2} [(λ_{1} - λ_{3}) sin θ_{r} + (λ_{2} - λ_{4}) cos θ_{r}] \\ q (t) = \frac{1}{2} [(λ_{2} - λ_{4}) sin θ_{r} - (λ_{1} - λ_{3}) cos θ_{r}] \end{matrix}, \end{array}

B (t) = [\begin{matrix} - sin θ_{1} & sin θ_{2} & sin θ_{3} & - sin θ_{4} \\ cos θ_{1} & - cos θ_{2} & cos θ_{3} & cos θ_{4} \\ λ_{1} & - λ_{2} & - λ_{3} & λ_{4} \end{matrix}], \begin{matrix} λ_{1} = l_{1} cos (θ_{1} - ϕ_{1}) \\ λ_{2} = l_{2} cos (θ_{2} - ϕ_{2}) \\ λ_{3} = l_{3} cos (θ_{3} - ϕ_{3}) \\ λ_{4} = l_{4} cos (θ_{4} - ϕ_{4}) \end{matrix}, and u_{r} (t) = [\begin{matrix} f_{r 1} \\ f_{r 2} \\ f_{r 3} \\ f_{r 4} \end{matrix}]

In order to more closely describe the motion of the walking assistance training robot when the user manipulate it, the dynamic model has considered the mass of the walking assistance training robot, m _r, and its user’s equivalent mass, m _u, in addition, $I^{'}$ and $m_{u} r_{0}^{2}$ are the inertial masses of the walking assistance training robot and the user, respectively. Moreover, $f_{i} (i = 1, 2, 3, 4)$ are the inputs, and $λ_{1}, λ_{2}, λ_{3}, λ_{4}$ and $r_{0}$ are the random parameters caused by the random shifts in the center of gravity. $θ_{r}$ is the angle between the first omniwheel and the $X^{'}$ -axis (i.e. $θ_{r} = θ_{1}$ ); thus, $θ_{2} = θ_{r} + \frac{π}{2}$ , $θ_{3} = θ_{r} + π$ , and $θ_{4} = θ_{r} + \frac{3 π}{2}$ .

The random shifts in the center of gravity has an internal system uncertainty, which tends to be problematic in many mechanical systems such as mobile robots and manipulators. In previous studies, the random shifts in the center of gravity was considered as a constant such that general control could be conducted. However, the proposed walking assistance training robot must work in narrow spaces such as indoor environments. Therefore, to ensure the safety of the user during rehabilitation, the walking assistance training robot requires a higher trajectory-tracking accuracy than most vehicles do. Thus, in a previous study,²⁴ the random shifts in the center of gravity were described using a stochastic model. Based on this constructed model, a controller was designed, and the developed trajectory-tracking system was shown to be stable. Applying the approach in the study by Chang et al.²⁴ and defining $S_{r} (t) = {\dot{X}}_{r} (t)$ , the stochastic model of random shifts in the center of gravity can be expressed as follows

d X_{r} (t) = S_{r} (t) d t

d S_{r} (t) = {\bar{M}}^{- 1} {\bar{B}}^{*} (t) u_{r} (t) d t + {\bar{M}}^{- 1} N Σ d w

where

\begin{array}{l} \bar{M} = [\begin{matrix} m_{r} + m_{u} & 0 & 0 \\ 0 & m_{r} + m_{u} & 0 \\ 0 & 0 & I^{'} \end{matrix}], {\bar{B}}^{*} (t) = [\begin{matrix} - sin θ_{1} & sin θ_{2} & sin θ_{3} & - sin θ_{4} \\ cos θ_{1} & - cos θ_{2} & cos θ_{3} & cos θ_{4} \\ L & L & L & L \end{matrix}], and \\ N = [\begin{matrix} - (m_{r} + m_{u}) sin θ_{1} & - {\dot{θ}}_{2}^{2} (m_{r} + m_{u}) cos θ_{2} & {\dot{θ}}_{3}^{2} (m_{r} + m_{u}) cos θ_{3} & (m_{r} + m_{u}) sin θ_{4} & 0 \\ (m_{r} + m_{u}) cos θ_{1} & - {\dot{θ}}_{2}^{2} (m_{r} + m_{u}) sin θ_{2} & {\dot{θ}}_{3}^{2} (m_{r} + m_{u}) sin θ_{3} & - (m_{r} + m_{u}) cos θ_{4} & 0 \\ 0 & 0 & 0 & 0 & 1 \end{matrix}] \end{array}

Here, $Σ$ is a nonnegative matrix and $w$ is a five-dimensional standard Wiener process.

In the stochastic model defined in equations (2) and (3), the time-varying arm of force in omniwheel was assumed to be the same as that in a standard wheel (i.e. the system was modeled without driven wheels); moreover, it was assumed that a touch force was exerted at the middle of each of the two rows of driven wheels and the arm of force was calculated relative to the geometric center to this point. However, the structure of the omniwheel and the actual time-varying arm of force in omniwheel are shown in Figures 3 and 4, respectively. In reality, the time-varying arm of force in omniwheel will shift among the two rows of driven wheels. In addition, since ${\bar{B}}^{*} (t)$ has changed to $B^{*} (t)$ , the inputs and states should also be changed to $u (t)$ and $X (t)$ , respectively. Thus, when the time-varying arm of force in omniwheel is considered a dynamic system, the stochastic model defined in equations (2) and (3) must be modified as follows

d X (t) = S (t) d t

d S (t) = {\bar{M}}^{- 1} B^{*} (t) u (t) d t + {\bar{M}}^{- 1} N Σ d w

where

X (t) = [\begin{matrix} x (t) \\ y (t) \\ θ (t) \end{matrix}], u (t) = [\begin{matrix} f_{1} \\ f_{2} \\ f_{3} \\ f_{4} \end{matrix}]

\begin{array}{l} B^{*} (t) = [\begin{matrix} - sin θ_{1} & sin θ_{2} & sin θ_{3} & - sin θ_{4} \\ cos θ_{1} & - cos θ_{2} & cos θ_{3} & cos θ_{4} \\ L + τ_{1} (t) & L + τ_{2} (t) & L + τ_{3} (t) & L + τ_{4} (t) \end{matrix}] \\ = [\begin{matrix} - sin θ_{1} & sin θ_{2} & sin θ_{3} & - sin θ_{4} \\ cos θ_{1} & - cos θ_{2} & cos θ_{3} & cos θ_{4} \\ L & L & L & L \end{matrix}] + [\begin{matrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ τ_{1} (t) & τ_{2} (t) & τ_{3} (t) & τ_{4} (t) \end{matrix}] \\ = {\bar{B}}^{*} (t) + B_{τ}^{*} (t) \end{array}

and $τ_{1} (t), τ_{2} (t), τ_{3} (t), and τ_{4} (t)$ are unknown functions. The improved dynamic stochastic model in equations (4) and (5) can be further simplified to

d X (t) = S (t) d t

d S (t) = {\bar{M}}^{- 1} {\bar{B}}^{*} (t) u (t) d t + Γ (t) d t + {\bar{M}}^{- 1} N Σ d w

where $Γ^{T} (t) = [γ_{1} (t), γ_{2} (t), γ_{3} (t)]$ , and $γ_{1} (t) = γ_{2} (t) = 0$ , $γ_{3} (t) = I^{'} \sum_{i = 1}^{4} τ_{i} (t) u_{i} (t)$ .

As equation (7) implies, the time-varying arm of force in omniwheel primarily influences the angle velocity, $\dot{θ}$ , and thereby affects the angle, $θ$ , of the walking assistance training robot.

Remark 1

Generally, as the matrix ${\bar{B}}^{*} (t)$ has many periodic functions that depend on $θ$ , the time-varying arm of force in omniwheel would influence the positions of X- and Y-axes at the same time. However, the controller that will be designed later has a decoupling function that can ideally eliminate this influence. Therefore, the time-varying arm of force in omniwheel amplifies the jitter in the position and velocity of walking assistance training robot angle described earlier.

Remark 2

Equations (7) and $γ_{3} (t) = I^{'} \sum_{i = 1}^{4} τ_{i} (t) u_{i} (t)$ show another conclusion that the scope of jitter is affected by the size of input. This expresses why previous studies can reduce the jitter caused by the time-varying arm of force in omniwheel by reducing the gain of the controller, as mentioned in introduction.

Assumption 1

Physically speaking, the angular velocity, $\dot{θ}$ , the mass of the robot m _r, and the mass of the user m _r + m _u are all bounded. Thus, there exists a $T^{'} > 0$ that satisfies the following inequality

{‖ {\bar{M}}^{- 1} ‖}_{F}^{2} [2 {\dot{θ}}^{4} {(m_{r} + m_{u})}^{2} + 2 {(m_{r} + m_{u})}^{2} + 1] {‖ Σ ‖}_{F}^{2} \leq T^{'}

where ${‖ • ‖}_{F}$ denotes the Frobenius norm.

Control design

Design of the velocity observer and tracking system

To estimate the unmeasurable statev(t), a velocity observer was designed as follows

d \bar{S} (t) = [{\bar{M}}^{- 1} {\bar{B}}^{*} (θ) u (t) + \hat{Γ} (t) - k \hat{S}] d t

\hat{S} (t) = \bar{S} (t) + k X (t)

where $\hat{S} (t) \in R^{n}$ is the estimate of $S (t)$ , $\bar{S} (t)$ is the internal state, and $k > 0$ is the gain (to be designed later). From equations (9) and (10), given the observer errors, $\tilde{S} (t) = S (t) - \hat{S} (t) = {[{\tilde{s}}_{1}, {\tilde{s}}_{2}, {\tilde{s}}_{3}]}^{T}$ , we have

d \hat{S} (t) = [{\bar{M}}^{- 1} {\bar{B}}^{*} (θ) u (t) + \hat{Γ} (t) + k \tilde{S} (t)] d t

d \tilde{S} (t) = - k \tilde{S} (t) d t + \tilde{Γ} (t) d t + {\bar{M}}^{- 1} N Σ d w

where $\tilde{Γ} (t) = Γ (t) - \hat{Γ} (t) = {[{\tilde{r}}_{1}, {\tilde{r}}_{2}, {\tilde{r}}_{3}]}^{T}$ and $\hat{Γ} (t) = {[{\hat{r}}_{1}, {\hat{r}}_{2}, {\hat{r}}_{3}]}^{T}$ . Here, we focus on the trajectory-tracking control so the tracking errors were designed as follows with respect to the reference trajectory, $X_{d} (t) \in C^{2} (R^{3})$

Z_{1} (t) = X (t) - X_{d} (t)

Z_{2} (t) = \hat{S} (t) - {\dot{X}}_{d} (t) - σ Z_{1} (t)

where $σ$ is a design parameter. By combining equations (6) and (7) with equations (13) and (14), respectively, the trajectory-tracking error system, ${(Z_{1}^{T} (t), Z_{2}^{T} (t))}^{T}$ , can be made available

d Z_{1} (t) = [Z_{2} (t) + σ Z_{1} (t) + \tilde{S} (t)] d t

d Z_{2} (t) = [{\bar{M}}^{- 1} {\bar{B}}^{*} (t) u (t) + \hat{Γ} (t) + (k - σ) \tilde{S} (t) - {\ddot{X}}_{d} (t) - σ Z_{2} (t) - σ^{2} Z_{1} (t)] d t

Design of the adaptive law and dynamic output feedback controller

The Lyapunov function is designed as follows

V (t) = \frac{1}{4} {(Z_{1}^{T} (t) Z_{1} (t))}^{2} + \frac{1}{4} {(Z_{2}^{T} (t) Z_{2} (t))}^{2} + \frac{1}{4} {({\tilde{S}}^{T} (t) \tilde{S} (t))}^{2} + \frac{1}{4} {({\tilde{Γ}}^{T} (t) \tilde{Γ} (t))}^{2}

The infinitesimal generator of equation (17) along with equations (15) and (16) satisfies

\begin{array}{l} L V (t) = Z_{1}^{T} (t) Z_{1} (t) Z_{1}^{T} (t) (Z_{2} (t) + σ Z_{1} (t) + \tilde{S} (t)) \\ + Z_{2}^{T} (t) Z_{2} (t) Z_{2}^{T} (t) ({\bar{M}}^{- 1} {\bar{B}}^{*} (t) u (t) + \hat{Γ} (t) + (k - σ) \tilde{S} (t) - {\ddot{X}}_{d} (t) - σ Z_{2} (t) - σ^{2} Z_{1} (t)) \\ + {\tilde{S}}^{T} (t) \tilde{S} (t) {\tilde{S}}^{T} (t) (- k \tilde{S} (t) + \tilde{Γ} (t)) + {\tilde{Γ}}^{T} (t) \tilde{Γ} (t) {\tilde{Γ}}^{T} (t) \dot{\tilde{Γ}} (t) \\ + \frac{1}{2} T r {Σ^{T} N^{T} {({\bar{M}}^{- 1})}^{T} (2 \tilde{S} (t) {\tilde{S}}^{T} (t) ​ ​ + ​ {\tilde{S}}^{T} (t) \tilde{S} (t) I) {\bar{M}}^{- 1} N Σ} \end{array}

By analyzing equation (18) and applying Young’s inequality and equation (8), the following inequalities can be obtained

Z_{1}^{T} (t) Z_{1} (t) Z_{1}^{T} (t) Z_{2} (t) \leq \frac{3}{4} ε_{1}^{\frac{4}{3}} {| Z_{1} (t) |}^{4} ​ ​ ​ + ​ \frac{1}{4 ε_{1}^{4}} {| Z_{2} (t) |}^{4}

Z_{1}^{T} (t) Z_{1} (t) Z_{1}^{T} (t) \tilde{S} (t) \leq \frac{3}{4} ε_{2}^{\frac{4}{3}} {| Z_{1} (t) |}^{4} + \frac{1}{4 ε_{2}^{4}} {| \tilde{S} (t) |}^{4}

(k - σ) Z_{2}^{T} (t) Z_{2} (t) Z_{2}^{T} (t) \tilde{S} (t) \leq \frac{3}{4} ε_{3}^{\frac{4}{3}} {(k - σ)}^{\frac{4}{3}} {| Z_{2} (t) |}^{4} + \frac{1}{4 ε_{3}^{4}} {| \tilde{S} (t) |}^{4}

{\tilde{S}}^{T} (t) \tilde{S} (t) {\tilde{S}}^{T} (t) \tilde{Γ} (t) \leq \frac{3}{4} ε_{4}^{\frac{4}{3}} {| \tilde{S} (t) |}^{4} + \frac{1}{4 ε_{4}^{4}} {| \tilde{Γ} (t) |}^{4}

\begin{array}{l} \frac{1}{2} T r {Σ^{T} N^{T} {({\bar{M}}^{- 1})}^{T} (2 \tilde{S} (t) {\tilde{S}}^{T} (t) + {\tilde{S}}^{T} (t) \tilde{S} (t) I) {\bar{M}}^{- 1} N Σ} \\ \leq \frac{3}{2} {\tilde{S}}^{T} (t) \tilde{S} (t) {‖ {\bar{M}}^{- 1} ‖}_{F}^{2} {‖ N ‖}_{F}^{2} {‖ Σ ‖}_{F}^{2} \\ \leq \frac{3}{2} {\tilde{S}}^{T} (t) \tilde{S} (t) {‖ {\bar{M}}^{- 1} ‖}_{F}^{2} [2 {\dot{θ}}^{4} {(m_{r} + m_{u})}^{2} + 2 {(m_{r} + m_{u})}^{2} + 1] {‖ Σ ‖}_{F}^{2} \\ \leq \frac{3}{2} {\tilde{S}}^{T} (t) \tilde{S} (t) T^{'} \leq \frac{9 ε_{5}^{2}}{4} {({\tilde{S}}^{T} (t) \tilde{S} (t))}^{2} + \frac{1}{2 ε_{5}^{2}} {T^{'}}^{2} \end{array}

where $ε_{1} > 0, ε_{2} > 0, ε_{3} > 0, ε_{4} > 0, and ε_{5} > 0$ are designed parameters.

Substituting equations (19) to (23) into equation (18), the following can be obtained

\begin{array}{l} L V (t) = Z_{1}^{T} (t) Z_{1} (t) Z_{1}^{T} (t) (\frac{3}{4} ε_{1}^{\frac{4}{3}} Z_{1} (t) + σ Z_{1} (t) + \frac{3}{4} ε_{2}^{\frac{4}{3}} Z_{1} (t)) \\ + Z_{2}^{T} (t) Z_{2} (t) Z_{2}^{T} (t) (\frac{1}{4 ε_{1}^{4}} Z_{2} (t) + {\bar{M}}^{- 1} {\bar{B}}^{*} (t) u (t) + \hat{Γ} (t) + \frac{3}{4} ε_{3}^{\frac{4}{3}} {(k - σ)}^{\frac{4}{3}} Z_{2} (t) - {\ddot{X}}_{d} - σ Z_{2} (t) - σ^{2} Z_{1} (t)) \\ + {\tilde{S}}^{T} (t) \tilde{S} (t) {\tilde{S}}^{T} (t) (\frac{1}{4 ε_{2}^{4}} \tilde{S} (t) + \frac{1}{4 ε_{3}^{4}} \tilde{S} (t) + \frac{3}{4} ε_{4}^{\frac{4}{3}} \tilde{S} (t) - k \tilde{S} (t) + \frac{9 ε_{5}^{2}}{4} \tilde{S} (t)) \\ + {\tilde{Γ}}^{T} (t) \tilde{Γ} (t) {\tilde{Γ}}^{T} (t) (\frac{1}{4 ε_{4}^{4}} \tilde{Γ} (t) + \dot{\tilde{Γ}} (t)) + \frac{1}{2 ε_{5}^{2}} {T^{'}}^{2} . \end{array}

Hence, the dynamic output feedback controller is designed as follows

\begin{array}{l} u (t) = {\bar{B}}^{* T} (θ) {({\bar{B}}^{*} (θ) {\bar{B}}^{* T} (θ))}^{- 1} \bar{M} \\ (- ξ_{2} Z_{2} (t) - \frac{1}{4 ε_{1}^{4}} Z_{2} (t) - \hat{Γ} (t) - \frac{3}{4} ε_{3}^{\frac{4}{3}} {(k - σ)}^{\frac{4}{3}} Z_{2} (t) + {\ddot{X}}_{d} (t) + σ Z_{2} (t) + σ^{2} Z_{1} (t)) \end{array}

where

σ = - ξ_{1} - \frac{3}{4} ε_{1}^{\frac{4}{3}} - \frac{3}{4} ε_{2}^{\frac{4}{3}}

k = ξ_{3} + \frac{1}{4 ε_{2}^{4}} + \frac{1}{4 ε_{3}^{4}} + \frac{3}{4} ε_{4}^{\frac{4}{3}} - \frac{9}{4} ε_{5}^{2}

Here, $ξ_{1} > 0, ξ_{2} > 0, and ξ_{3} > 0$ are the design parameters.

Then, the adaptive laws are designed as follows

\begin{array}{l} \dot{\hat{Γ}} (t) = - (\frac{1}{4 ε_{4}^{4}} + ξ_{4}) \hat{Γ} (t) + \ddot{\hat{S}} (t) \\ - {\ddot{S}}_{r} (t) - {\bar{M}}^{- 1} {\dot{\bar{B}}}^{*} (t) (u (t) - u_{r} (t)) \\ - {\bar{M}}^{- 1} {\bar{B}}^{*} (t) (\dot{u} (t) - {\dot{u}}_{r} (t)) \\ + (\frac{1}{4 ε_{4}^{4}} + ξ_{4}) (\dot{\hat{S}} (t) - {\dot{S}}_{r} (t) - {\bar{M}}^{- 1} {\bar{B}}^{*} (t) (u (t) - u_{r} (t))) \end{array}

where $ξ_{4} > 0$ is a design parameter.

Combining equations (5), (7), and (28), yields

\frac{1}{4 ε_{4}^{4}} \tilde{Γ} (t) + \dot{\tilde{Γ}} (t) = - ξ_{4} \tilde{Γ} (t)

Substituting equations (25) to (29) into equation (24), we have

\begin{array}{l} L V (t) \leq - ξ_{1} {(Z_{1}^{T} (t) Z_{1} (t))}^{2} - ξ_{2} {(Z_{2}^{T} (t) Z_{2} (t))}^{2} \\ - ξ_{3} {({\tilde{S}}^{T} (t) \tilde{S} (t))}^{2} - ξ_{4} {({\tilde{Γ}}^{T} (t) \tilde{Γ} (t))}^{2} + \frac{1}{2 ε_{5}^{2}} {T^{'}}^{2} \\ \leq - ξ V (t) + ϖ \end{array}

where $ξ = \frac{1}{4} min {ξ_{1}, ξ_{2}, ξ_{3}, ξ_{4}}$ and $ϖ = \frac{1}{2 ε_{5}^{2}} {T^{'}}^{2}$ .

Remark 3

Equation (3) is an exceptional situation of equation (7) when $Γ = 0$ . Therefore, $u_{r} (t)$ in equation (28) would be designed to have the same structure as equation (25) but the parameters are redefined and $\hat{Γ} = 0$

\begin{array}{l} u_{r} (t) = {\bar{B}}^{* T} (θ) {({\bar{B}}^{*} (θ) {\bar{B}}^{* T} (θ))}^{- 1} {\bar{M}}^{- 1} \\ (- ξ_{r 2} Z_{r 2} (t) - \frac{1}{4 ε_{r 1}^{4}} Z_{r 2} (t) - \frac{3}{4} ε_{r 3}^{\frac{4}{3}} σ^{\frac{4}{3}} Z_{r 2} (t) + {\ddot{X}}_{d} (t) + σ Z_{r 2} (t) + σ^{2} Z_{r 1} (t)) \end{array}

where $Z_{r 1} (t) = X_{r} (t) - X_{d} (t)$ and $Z_{r 2} (t) = S_{r} (t) - {\dot{X}}_{d} (t) - σ Z_{r 1} (t)$ .

Analysis of stability

Theorem 1

For the stochastic dynamic model of the walking assistance training robot that can handle random shifts in the center of gravity and time-varying arm of force in omniwheel, the dynamic output feedback controller described in equation (25) is designed to track a reference trajectory, $X_{d} (t) \in C^{2} (R^{3})$ . Then, the controller (equation (25)) and the adaptive laws (equation (28)) guarantee the error system, ${(Z_{1}^{T} (t), Z_{2}^{T} (t), {\tilde{S}}^{T} (t), {\tilde{Γ}}^{T} (t))}^{T}$ , and have a unique strong solution in the interval $[t_{0}, \infty)$ . In addition, exponentially practical stability of ${(Z_{1}^{T} (t), Z_{2}^{T} (t), {\tilde{S}}^{T} (t), {\tilde{Γ}}^{T} (t))}^{T}$ is achieved for the initial values, $Z_{1} (t_{0}) \in R^{3}$ , $Z_{2} (t_{0}) \in R^{3}$ , $\tilde{S} (t_{0}) \in R^{3}$ , and $\tilde{Γ} (t_{0}) \in R^{3}$ . Moreover, the tracking errors, $Z_{1} (t)$ and ${\dot{Z}}_{1} (t)$ , the observer error, $\tilde{Γ} (t)$ and $\tilde{S} (t)$ satisfy the following

lim_{t \to \infty} E | Z_{1} (t) | \leq \sqrt{2} (\frac{ϖ}{ξ})^{\frac{1}{4}}

lim_{t \to \infty} E | {\dot{Z}}_{1} (t) | \leq 2 \sqrt{2} (1 + | σ |) (\frac{ϖ}{ξ})^{\frac{1}{4}}

lim_{t \to \infty} E | \tilde{S} (t) | \leq \sqrt{2} (\frac{ϖ}{ξ})^{\frac{1}{4}}

lim_{t \to \infty} E | \tilde{Γ} (t) | \leq \sqrt{2} (\frac{ϖ}{ξ})^{\frac{1}{4}}

It is noteworthy that the terms on the right-hand sides of equations (32) to (35) could be adjusted to sufficient small by selecting an appropriate set of design parameters.

Proof

In the stochastic dynamic model of the walking assistance training robot, B(t), and N(t) are smooth matrixes, which insure the local Lipschitz condition of them hold. In the same way, it is easy to receive that the local Lipschitz conditions of u (t) and $Γ (t)$ also hold. Thus, the error system, ${(Z_{1}^{T} (t), Z_{2}^{T} (t), {\tilde{S}}^{T} (t), {\tilde{Γ}}^{T} (t))}^{T}$ , would satisfy the local Lipschitz condition. Based on Lemma 1 in the study by Cui et al.²⁶ and combining equations (17) and (30), a unique strong solution to ${(Z_{1}^{T} (t), Z_{2}^{T} (t), {\tilde{S}}^{T} (t), {\tilde{Γ}}^{T} (t))}^{T}$ exists in the interval [t ₀, ∞) with initial values of $Z_{1} (t_{0}) \in R^{n}$ , $Z_{2} (t_{0}) \in R^{3}$ , ${\tilde{S}}^{T} (t_{0}) \in R^{3}$ , and ${\tilde{Γ}}^{T} (t_{0}) \in R^{3}$ . Thus, the error system, ${(Z_{1}^{T} (t), Z_{2}^{T} (t), {\tilde{S}}^{T} (t), {\tilde{Γ}}^{T} (t))}^{T}$ , is exponentially practically stable.

Furthermore, the inequality in equation (30) can be multiplied by $e^{ξ t} > 0$ to find that

e^{ξ t} (L V (t) + ξ V (t)) \leq e^{ξ t} ϖ

Based on Lemma 3.3.1 in the study by Khasminskii²⁷ and integrating equation (36) from $t_{0}$ to $t$ , we have

E (e^{ξ t} V (t)) \leq e^{ξ t_{0}} V (t_{0}) + E \int_{t_{0}}^{t} e^{ξ s} ϖ \cdot d s \forall t \geq t_{0}

From equation (17), it can be found that

E | Z_{1} (t) | \leq e^{\frac{ξ}{4} (t_{0} - t)} (| Z_{1} (t_{0}) | + | Z_{2} (t_{0}) | + | \tilde{S} (t_{0}) | + | \tilde{Γ} (t_{0}) |) + \sqrt{2} (\frac{ϖ}{ξ})^{\frac{1}{4}}

E | Z_{2} (t) | \leq e^{\frac{ξ}{4} (t_{0} - t)} (| Z_{1} (t_{0}) | + | Z_{2} (t_{0}) | + | \tilde{S} (t_{0}) | + | \tilde{Γ} (t_{0}) |) + \sqrt{2} (\frac{ϖ}{ξ})^{\frac{1}{4}}

E | \tilde{S} (t) | \leq e^{\frac{ξ}{4} (t_{0} - t)} (| Z_{1} (t_{0}) | + | Z_{2} (t_{0}) | + | \tilde{S} (t_{0}) | + | \tilde{Γ} (t_{0}) |) + \sqrt{2} (\frac{ϖ}{ξ})^{\frac{1}{4}}

E | \tilde{Γ} (t) | \leq e^{\frac{ξ}{4} (t_{0} - t)} (| Z_{1} (t_{0}) | + | Z_{2} (t_{0}) | + | \tilde{S} (t_{0}) | + | \tilde{Γ} (t_{0}) |) + \sqrt{2} (\frac{ϖ}{ξ})^{\frac{1}{4}}

Considering equations (38) and (39) and given that $| {\dot{Z}}_{1} (t) | \leq | Z_{2} (t) | + | σ Z_{1} (t) | \leq (1 + | σ |) (| Z_{2} (t) | + | Z_{1} (t) |)$ , equations (32) to (35) hold. Noting that $ϖ = \frac{1}{2 ε_{5}^{2}} {T^{'}}^{2}$ , the right-hand sides of equations (32) to (35) can be adjusted to sufficient small by choosing appropriate values of $ξ$ and $ϖ$ , as they are independent of each other.

Remark 4

Applying Markov’s inequality and considering that $E {| Z_{1} (t) |}^{2} < \infty$ , $E {| {\dot{Z}}_{1} (t) |}^{2} < \infty$ , $E {| \tilde{S} (t) |}^{2} < \infty$ , and $E {| \tilde{Γ} (t) |}^{2} < \infty$ , For any set of values $ϑ_{1} > 0, ϑ_{2} > 0, ϑ_{3} > 0, and ϑ_{4} > 0$ , the following hold

P {| Z_{1} (t) | > ϑ_{1}} \leq \frac{1}{ϑ_{1}^{2}} E {| Z_{1} (t) |}^{2}

P {| {\dot{Z}}_{1} (t) | > ϑ_{2}} \leq \frac{1}{ϑ_{2}^{2}} E {| {\dot{Z}}_{1} (t) |}^{2}

P {| \tilde{S} (t) | > ϑ_{3}} \leq \frac{1}{ϑ_{3}^{2}} E {| \tilde{S} (t) |}^{2}

P {| \tilde{Γ} (t) | > ϑ_{4}} \leq \frac{1}{ϑ_{4}^{2}} E {| \tilde{Γ} (t) |}^{2}

Substituting equations (42) to (45) into equations (32) to (35), respectively, the following equations are obtained

lim_{t \to \infty} P {| Z_{1} (t) | > ϑ_{1}} \leq \frac{2}{ϑ_{1}^{2}} {(\frac{ϖ}{ξ})}^{\frac{1}{2}}

lim_{t \to \infty} P {| {\dot{Z}}_{1} (t) | > ϑ_{2}} \leq \frac{16}{ϑ_{2}^{2}} (1 + σ^{2}) {(\frac{ϖ}{ξ})}^{\frac{1}{2}}

lim_{t \to \infty} P {| \tilde{S} (t) | > ϑ_{3}} \leq \frac{2}{ϑ_{3}^{2}} {(\frac{ϖ}{ξ})}^{\frac{1}{2}}

lim_{t \to \infty} P {| \tilde{Γ} (t) | > ϑ_{4}} \leq \frac{2}{ϑ_{4}^{2}} {(\frac{ϖ}{ξ})}^{\frac{1}{2}}

By choosing the parameters in the controller and adaptive laws, the right-hand-side terms of equations (46) to (49) can be made sufficiently small, implying that asymptotical tracking with the controller in equation (25) can be realized with a given probability. Thus, when selecting the design parameters of the controller and the adaptive law, the effects of the tracking errors must be considered carefully.

Simulation study

Next, a simulation was conducted to show that the proposed method can be used to provide sufficient walking assistance training robot performance with random shifts in the center of gravity and time-varying arm of force in omniwheel. To verify the tracking performance of the walking assistance training robot with the proposed method, an astroid trajectory, $X_{d}$ , was set for the walking assistance training robot

\begin{matrix} x_{d} = 20 {cos}^{3} (0.1 t) \\ y_{d} = 20 {sin}^{2} (0.1 t) \\ θ_{d} = \frac{π}{4} \end{matrix}, 0 \leq t \leq 60 s

The physical parameters were $m_{r} = 58 kg$ , m _u = 60 kg, $I^{'} = 27.7 kg \cdot m^{2}$ , and $L = 0.4 m$ . The random parameters caused by random shifts in the center of gravity are $λ_{1} = L - 0.16 \cdot r a n d 1 m$ , $λ_{2} = L + 0.16 \cdot r a n d 2 m$ , $λ_{3} = L + 0.16 \cdot r a n d 1 m$ , $λ_{4} = L - 0.16 \cdot r a n d 2 m$ , and $r_{0} = \sqrt{{(r a n d 1)}^{2} + {(r a n d 2)}^{2}}$ m, where $0 \leq r a n d 1 \leq 1$ and $0 \leq r a n d 2 \leq 1$ are two groups of stochastic parameters. The design parameters of the controller are $ξ_{1} = 4.5$ , $ξ_{2} = 7.0$ , $ξ_{3} = 6.5$ , $ξ_{4} = 5.5$ , $ε_{1} = 1.4$ , $ε_{2} = 1.7$ , $ε_{3} = 1.9$ , $ε_{4} = 0.7$ , and $ε_{5} = 0.9$ . The dynamic model and controller are simulated with Microsoft Visual C++ 2010. Figure 5 shows the simulated variation in the arm of force. Figures 6 to 8 show the simulation results.

In Figure 7, the blue line is the reference trajectory, and the red line is the response of the walking assistance training robot on its x-axis component, y-axis component, orientation angle, and specific movement. Figure 7 shows the position trajectories in terms of the X-position, Y-position, and angle, as well as the astroid path. The simulation result shows that the walking assistance training robot can accurately track the astroid trajectory (equation (50)), indicating that exponentially practical stability of the trajectory-tracking system was obtained. Figure 8 shows the velocity tracking in terms of the X-position, Y-position, and angle. These results show that the velocity tracing was very accurate. The velocity observer (green line) was shown to accurately observe the velocity of the walking assistance training robot (red line). Figure 8(d) shows that the mean absolute errors were sufficiently small. The trajectory tracking was accurate not only because the velocity observer works effectively and the tracking controller can account for the random parameters resulting from the random shifts in the center of gravity but also the adaptive laws shown in Figure 6. The adaptive laws can immediately adjust the parameters in the controller to handle unknown functions in the dynamic model due to the time-varying arm of force in omniwheel.

To demonstrate the need for the time-varying arm of force in omniwheel analysis for the walking assistance training robot, a simulation was conducted with a proportional–integral–derivative (PID) controller for comparison. The physical parameters and the designed trajectory were the same as mentioned above but the center of gravity did not shift (i.e. $λ_{1} = λ_{2} = λ_{3} = λ_{4} = L$ m and $r_{0} = 0$ m). The PID control parameters were designed as $Kp = diag {5 .2, 7 .5, 9 .1}$ , $Ki = diag {0 .5, 0 .1, 0 .9}$ , and $Kd = diag {0 .3, 0 .7, 0.4}$ . Figures 9 and 10 show the simulation results.

Figures 9(a) and (b) and 10(a) and (b) depict the trajectory tracking on the X- and Y-axes was very accurate. However, Figures 9(c) and 10(c) show that there are tracking errors for the position and velocity angles fluctuated significantly over the entire simulation. Thus, it can be concluded that the PID controller don’t have the ability to effectively eliminate the jitter caused by the time-varying arm of force in omniwheel. This simulation showed that the time-varying arm of force in omniwheel significantly affects the angle in the trajectory tracking as indicated by the analysis in Remark 1, demonstrating the need to consider the time-varying arm of force in omniwheel. These simulation results support Theorem 1, as well as the effectiveness of the proposed method.

In order to better illustrate that the controller has good tracking ability, we give another group of simulation that makes the walking assistance training robot track a cycloid trajectory. The random parameters caused by random shifts in the center of gravity and the design parameters of the controller are same as mentioned above. The cycloid trajectory and simulation results are shown as follows

\begin{array}{l} x_{d} = 3 (0.3 t - sin (0.3 t)) \\ y_{d} = 10 (0.3 - cos (0.3 t)) \\ θ_{d} = \frac{π}{4} \end{array}, 0 \leq t \leq 60 s

Figure 11 gives the adaptive laws, in which ${\hat{γ}}_{3}$ is used to eliminate the jitter in orientation angle that is caused by the time-varying arm of force in omniwheel. In Figure 12, the position respond of walking assistance training robot (red line) can approach the reference trajectory (blue line), which illustrates that the position tracking on the X-axis, the Y-axis, orientation angle, as well as the cycloid path is achieved. In Figure 13, the velocity respond of walking assistance training robot (red line) can approach the velocity of reference trajectory (blue line), which illustrate that the velocity tracking in terms of the X-axis, the Y-axis, and orientation angle is realized. The observation of velocity observer (green line) and the velocity respond of walking assistance training robot (red line) are overlapped closely, which confirms that the velocity observation is accurate. Figure 13(d) shows that the mean absolute errors were sufficiently small.

Conclusions

In this study, we developed tracking control for a walking assistance training robot to handle random shifts in the center of gravity and time-varying arm of force in omniwheel using a stochastic dynamic model. Rather than using a velocity sensor, a velocity observer was designed to observe the velocity of the walking assistance training robot, which was combined the position information to form the input information for the controller. The analysis of time-varying arm of force in omniwheel revealed that it mainly influenced the jitter in orientation angle and the scope of jitter is affected by the size of input. An adaptive law was constructed to eliminate the vibrations caused by the time-varying arm of force in omniwheel. Then, using the Lyapunov function and Markov’s inequality, an output feedback trajectory-tracking controller was proposed. It was confirmed that the mean absolute errors in the trajectory tracking of the proposed walking assistance training robot were exponentially practically stable under both random shifts in the center of gravity and time-varying arm of force in omniwheel. In addition, the mean absolute error in the trajectory tracking was shown to become arbitrarily small after turning the design parameters. Moreover, the simulation results illustrate the effectiveness of the proposed scheme.

The proposed stochastic model, adaptive law, and output feedback trajectory-tracking controller are expected to be extensible to other mechanical systems that experience random shifts in the center of gravity and time-varying arm of force in omniwheel. In addition, omniwheels are more vulnerable than general wheels TOR friction effect. Therefore, considering both time-varying arm of force and friction, it will be more beneficial to the application of omniwheel.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was partly supported by the JSPS KAKENHI (grant no. 15H03951), the CANON Foundation, and the CASIO Science Promotion Foundation.

ORCID iD

Hongbin Chang

Shuoyu Wang

Ping Sun

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